Local and superlinear convergence of quasi-Newton methods based on modified secant conditions. (English) Zbl 1124.65054
The authors consider the question how to improve the secant condition which is imposed in quasi-Newton methods. They incorporate a relaxation parameter into the secant condition. The aim of this parameter is to smoothly switch the standard secant condition and the secant condition of J. Z. Zhang, N. Y. Deng and L. H. Chen [J. Optimization Theory Appl. 102, 147–167 (1999; Zbl 0991.90135)] and J. Z. Zhang and C. Xu [J. Comput. Appl. Math. 137, 269–278 (2001; Zbl 1001.65065)].
The authors consider a Broyden family that satisfies such a modified secant condition, which includes the BFGS-like and DFP-like updates. The introduced parameter enables to handle preserving positive definiteness of the approximation matrix more easily than the one based on the modified secant condition proposed by Zhang et al. [loc. cit.] The local and q-superlinear convergence of the presented method is proved.
The authors consider a Broyden family that satisfies such a modified secant condition, which includes the BFGS-like and DFP-like updates. The introduced parameter enables to handle preserving positive definiteness of the approximation matrix more easily than the one based on the modified secant condition proposed by Zhang et al. [loc. cit.] The local and q-superlinear convergence of the presented method is proved.
Reviewer: Nada Djuranović-Miličić (Belgrade)
MSC:
| 65K05 | Numerical mathematical programming methods |
| 90C30 | Nonlinear programming |
| 90C53 | Methods of quasi-Newton type |
Keywords:
unconstrained minimization; quasi-Newton method; modified secant condition; Broyden family; local and \(q\)-superlinear convergence; BFGS update; DFP updateReferences:
| [1] | Broyden, C. G.; Dennis, J. E.; Moré, J. J., On the local and superlinear convergence of quasi-Newton methods, J. Inst. Math. Appl., 12, 223-245 (1973) · Zbl 0282.65041 |
| [2] | Byrd, R. H.; Nocedal, J.; Yuan, Y., Global convergence of a class of quasi-Newton methods on convex problems, SIAM J. Numer. Anal., 24, 1171-1190 (1987) · Zbl 0657.65083 |
| [3] | Dennis, J. E.; Moré, J. J., A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comput., 28, 549-560 (1974) · Zbl 0282.65042 |
| [4] | J.E. Dennis Jr., R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, NJ, 1983.; J.E. Dennis Jr., R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, NJ, 1983. · Zbl 0579.65058 |
| [5] | Ford, J. A.; Moghrabi, I. A., Multi-step quasi-Newton methods for optimization, J. Comput. Appl. Math., 50, 305-323 (1994) · Zbl 0807.65062 |
| [6] | Ford, J. A.; Moghrabi, I. A., Using function-values multi-step quasi-Newton methods, J. Comput. Appl. Math., 66, 201-211 (1996) · Zbl 0856.65073 |
| [7] | Huang, H. Y., Unified approach to quadratically convergent algorithms for function minimization, J. Optim. Theory Appl., 5, 405-423 (1970) · Zbl 0184.20202 |
| [8] | Powell, M. J.D., How bad are the BFGS and DFP methods when the objective function is quadratic?, Math. Programming, 34, 34-47 (1986) · Zbl 0581.90068 |
| [9] | Stachurski, A., Superlinear convergence of Broyden’s bounded \(\theta \)-class of methods, Math. Programming, 20, 196-212 (1981) · Zbl 0448.90048 |
| [10] | Yabe, H.; Yamaki, N., Some properties of Oren’s SSVM type algorithm for unconstrained minimization, TRU Mathematics, 16, 103-111 (1980) · Zbl 0501.90075 |
| [11] | Yabe, H.; Yamaki, N., A bounded deterioration property of a symmetric positive definite class of Newton-like methods and its application, J. Oper. Res. Soc. of Japan, 42, 373-388 (1999) · Zbl 0998.90089 |
| [12] | Yamaki, N.; Yabe, H., A family of the quasi-Newton methods, TRU Math., 16, 49-54 (1980) · Zbl 0461.65053 |
| [13] | Zhang, J. Z.; Deng, N. Y.; Chen, L. H., New quasi-Newton equation and related methods for unconstrained optimization, J. Optim. Theory Appl., 102, 147-167 (1999) · Zbl 0991.90135 |
| [14] | Zhang, Y.; Tewarson, R. P., Quasi-Newton algorithms with updates from the preconvex part of Broyden’s family, IMA J. Numer. Anal., 8, 487-509 (1988) · Zbl 0661.65061 |
| [15] | Zhang, J. Z.; Xu, C. X., Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations, J. Comput. Appl. Math., 137, 269-278 (2001) · Zbl 1001.65065 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
